Mathematics

Problem

Let a data set have arithmetic mean

\bar{x}

and variance

\sigma^2

. a. (i) If each observation is increased by a non–zero constant

k

, show that the variance of the new data remains

\sigma^2

. b. (ii) If each observation is multiplied by a non–zero constant

m

, show that the variance of the new data becomes

m^2\sigma^2

. c. (iii) Verify the conclusions of parts (i) and (ii) for the observations 3, 5, 7, 10 by taking

k = 4

and

m = 2

. Calculate all variances explicitly.